REMARKS ON H*infin; CONTROLLER DESIGN FOR SISO PLANTS WITH TIME DELAYS 1

REMARKS ON H*infin; CONTROLLER DESIGN FOR SISO PLANTS WITH TIME DELAYS 1

Preprints of the 5th IFAC Symposium on Robust Control Design ROCOND'06, Toulouse, France, July 5-7, 2006 REMARKS ON H∞ CONTROLLER DESIGN FOR SISO PL...

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Preprints of the 5th IFAC Symposium on Robust Control Design

ROCOND'06, Toulouse, France, July 5-7, 2006

REMARKS ON H∞ CONTROLLER DESIGN FOR SISO PLANTS WITH TIME DELAYS 1 ∗∗ ¨ Suat G¨ um¨ u¸ssoy ∗ Hitay Ozbay



was with Dept. of Electrical and Computer Eng., Ohio State University, Columbus, OH 43210, U.S.A.; current affiliation: MIKES Inc., Akyurt, Ankara TR-06750, Turkey, [email protected] ∗∗ Dept. of Electrical and Electronics Eng., Bilkent University, Bilkent, Ankara TR-06800, Turkey, on leave from Dept. of Electrical and Computer Eng., Ohio State University, Columbus, OH 43210, U.S.A., [email protected], [email protected]

Abstract: The skew Toeplitz approach is one of the well developed methods to design H∞ controllers for infinite dimensional systems. In order to be able to use this method the plant needs to be factorized in some special manner. This paper investigates the largest class of SISO time delay systems for which the special factorizations required by the skew Toeplitz approach can be done. Reliable implementation of the optimal controller is also discussed. It is shown that the finite impulse response (FIR) block structure appears in these controllers not only for plants with I/O delays, but also for general time-delay plants. Keywords: H∞ control, time-delay, mixed sensitivity problem

1. INTRODUCTION There are many well-developed techniques for finding H∞ optimal and suboptimal controllers for systems with time delays. In particular, when the plant is a dead-time system: e−hs P0 (s) where P0 is a rational SISO plant, the optimal H∞ control problem is solved by (Zhou and Khargonekar, 1987), (Foias et al., 1986), using operator theoretic methods; see ¨ also (Smith, 1989), (Ozbay, 1990) and their references. State-space solution to the same problem is given in (Tadmor, 1997), and (Meinsma and Zwart, 2000). Notably, (Meinsma and Zwart, 2000) used J-spectral factorization approach to solve the MIMO version of the problem. Moreover they showed the finite impulse response (FIR) structure appearing in the reliable im1 This work was supported in part by the European Commission ¨ ITAK ˙ (contract no. MIRG-CT-2004-006666) and by TUB (grant no. EEEAG-105E065).

plementation of the H∞ controllers for dead-time systems. (Meinsma and Mirkin, 2005) extended this result to the multi-delay dead-time systems (input/output delay case). A closed-form controller formula is obtained by (Kashima and Yamamoto, 2003) for the sensitivity minimization problem involving pseudorational plants. For more general infinite dimensional plants a solution is given by (Foias et al., 1996). Their approach needs inner-outer factorization of the plant. (Toker and ¨ Ozbay, 1995) simplified this method and brought into a compact form. (Kashima, 2005) obtained an expression for the optimal H∞ controller for the plants that can be expressed as a cascade connection of a finite-dimensional generalized plant and a scalar inner function. As it was done by (Mirkin, 2003), the solution is reduced to solving two algebraic Riccati equations and an additional one-block

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problem. Moreover, (Kashima, 2005) gave the innerouter factorizations of stable pseudorational systems. In our study, we determine the largest class of timedelay systems (TDS) for which the Skew-Toeplitz approach of (Foias et al., 1996) is applicable. In order to use this method it is necessary to do inner-outer factorizations of the plant. An additional assumption is that the infinite dimensional plant has finitely many unstable zeros or poles. In this paper, we give necessary and sufficient conditions for TDS to have finitely many unstable zeros or poles. We classify the TDS and give conditions such that the desired factorization is possible. For admissible plants, the factorization is given and optimal H∞ controller is obtained. The unstable polezero cancellation in the optimal controller expression ¨ of (Toker and Ozbay, 1995) is eliminated. This way ¨ we establish the link between (Toker and Ozbay, 1995) and (Meinsma and Zwart, 2000) by showing the FIR structure appearing in H∞ controllers for not only dead-time plants, but also for more general TDS. 2. PRELIMINARY DEFINITIONS AND RESULTS ¨ In (Foias et al., 1996; Toker and Ozbay, 1995), it is assumed that the plant is in the form ˆo (s) m ˆ n (s)N Pˆ (s) = (1) m ˆ d (s) where m ˆ n (s) is inner, infinite dimensional and m ˆ d (s) ˆo (s) is outer, possiis inner, finite dimensional and N bly infinite dimensional. The optimal H∞ controller, Cˆopt , stabilizes the feedback system and achieves the minimum H∞ cost, γˆopt :   ˆ 1 (1 + Pˆ Cˆopt )−1   W   γˆopt =  ˆ ˆ ˆ (2) −1  ˆ ˆ W2 P Copt (1 + P Copt ) ∞

ˆ 1 and W ˆ 2 are finite dimensional weights of the where W mixed sensitivity minimization problem. Recently, the optimal H∞ control problem is solved ¨ by (G¨ um¨ u¸ssoy and Ozbay, 2004) for systems with infinitely many unstable poles and finitely many unstable zeros by using the duality with the problem (2). In this case, the plant has a factorization ˜o (s) m ˜ d (s)N P˜ (s) = (3) m ˜ n (s) where m ˜ n is inner, infinite dimensional, m ˜ d (s) is finite ˜o (s) is outer, possibly infinite dimensional, inner, and N dimensional. For this dual problem, the optimal controller, C˜opt , and minimum H∞ cost, γ˜opt , are found for the mixed problem sensitivity minimization  ˜ 1 (1 + P˜ C˜opt )−1   W  γ˜opt =  (4)  W ˜ 2 P˜ C˜opt (1 + P˜ C˜opt )−1  . ∞

In this paper we consider systems: n general−hdelay rp,i (s)e i s rp (s) i=1 = m (5) P (s) = −τj s tp (s) j=1 tp,j (s)e satisfying the assumptions A.1 (a) rp,i (s) and tp,j (s) are polynomials with real coefficients; (b) hi , τj are rational numbers such that 0 ≤ h1 < h2 < . . . < hn , and 0 ≤ τ1 < τ2 < . . . < τm , with h1 ≥ τ1 ;

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(c) define the polynomials rp,imax and tp,jmax with largest polynomial degree in rp,i and tp,j respectively (the smallest index if there is more than one), then, deg{rp,imax (s)} ≤ deg{tp,jmax (s)} and himax ≥ τjmax where deg{.} denotes the degree of the polynomial; A.2 P has no imaginary axis zeros or poles; A.3 P has finitely many unstable poles or zeros, or equivalently rp (s) or tp (s) has finitely many zeros in C+ ; A.4 P can be written in the form of (1) or (3). Conditions stated in A.1 are not restrictive. In most cases A.2 can be removed if the weights are chosen in a special manner. The conditions A.3 − A.4 come from the Skew-Toeplitz approach. It is not easy to check assumptions A.3−A.4, unless a quasi-polynomial root finding algorithm is used. We will give a necessary and sufficient condition to check the assumption A.3 in section 2.1 and give conditions to check the assumption A.4 in section 3.1. By simple rearrangement, P can be written as, n R (s)e−hi s R(s) i P (s) = = i=1 (6) m −τj s T (s) j=1 Tj (s)e where Ri and Tj are finite dimensional, stable, proper transfer functions. The assumptions A.1−A.4 and rearrangement of the plant are illustrated on the following example. Consider the system x˙ 1 (t) = −x1 (t − 0.2) − x2 (t) + u(t) + 2u(t − 0.4), x˙ 2 (t) = 5x1 (t − 0.5) − 3u(t) + 2u(t − 0.4), y(t) = x1 (t). (7) whose transfer function is in the form 2 rp,i (s)e−hi s rp (s) = i=1 , P (s) = 3 −τi s tp (s) i=1 tp,i (s)e (s + 3)e−0s + 2(s − 1)e−0.4s . (8) s2 e−0s + se−0.2s + 5e−0.5s Note that rp,i and tp,j are polynomials with real coefficients, delays are nonnegative with increasing order. By imax = 1 and jmax = 1, h1 = 0 ≥ τ1 = 0 and deg{rp,1 (s)} = 1 ≤ deg{tp,1 (s)} = 2. Therefore, assumption A.1 is satisfied. The plant, P has no imaginary axis poles or zeros (assumption A.2). The denominator of the plant, tp (s) has finitely many unstable zeros at 0.4672 ± 1.8890j, whereas rp (s) has infinitely many unstable zeros converging to 1.7329 − (5k + 2.5)πj as k → ∞. Therefore, plant has finitely many unstable poles satisfying assumption A.3. One can show that the plant can be factorized as (1). In this example we have 2 Ri (s)e−hi s R = i=1 (9) P = 3 −τi s T i=1 Tj (s)e where rp,i (s) tp,j (s) , and Tj = Ri (s) = (s + 1)2 (s + 1)2 are stable proper finite dimensional transfer functions. Below we give conditions such that A.3 − A.4 can be checked easily. =

Preprints of the 5th IFAC Symposium on Robust Control Design

2.1 Time Delay Systems with Finitely Many Unstable Zeros or Poles n Definition 2.1. Consider R(s) = i=1 Ri (s)e−hi s where each Ri is a rational, proper, stable transfer function with real coefficient, and 0 ≤ h1 < h2 < . . . < hn . Let relative degree of Ri (s) be di , then (i) if d1 < max {d2 , . . . , dn }, then R(s) is a retardedtype time-delay system (RTDS), (ii) if d1 = max {d2 , . . . , dn }, then R(s) is a neutraltype time-delay system (NTDS), (iii) if d1 > max {d2 , . . . , dn }, then R(s) is an advanced-type time-delay system (ATDS). Note that if R and T are ATDS, plant has always infinitely many unstable zeros and poles which is not a valid plant for Skew-Toeplitz approach. It is wellknown that RTDS has finitely unstable zeros on the right-half plane, (Bellman and Cooke, 1963). Therefore, we will give a necessary and sufficient condition to check whether a NTDS has finitely many or infinitely many unstable zeros with the following lemma: Lemma 2.1. Assume that R(s) is a NTDS with no imaginary axis zeros and poles, then the system, R, has finitely many unstable zeros if and only if all the  ˜ ˜ roots of the polynomial, ϕ(r) = 1 + ni=2 ξi rhi −h1 has magnitude greater than 1 where ξi = lim Ri (jω)R1−1 (jω) ∀ i = 2, . . . , n, ω→∞

˜i h ˜ i ∈ Z+ , ∀ i = 1, . . . , n. , N, h hi = N Proof. Since delays are rational numbers, there exist ˜ i . If R is a NTDS, there is no positive integers N and h root with real part extending to infinity, i.e.,  n   R(s)eh1 s    ≥ 1 − lim |ξi |e−(hi −h1 )σ > 0  R1 (s)  σ→∞ σ→∞

i=2

where s = σ+jω. Therefore, NTDS may have infinitely many unstable zeros extending to infinity in imaginary part with bounded positive real part, see (Bellman and Cooke, 1963). R has finitely many unstable zeros if and only if R(σ + jω) has finitely many zeros as ω → ∞ and 0 < σ < σo < ∞. Equivalently, R(σ + jω) has finitely many unstable zeros if and only if n   R(s) ˜ ˜  lim = 1 + ξi rhi −h1 (10) ω→∞ R1 (s)e−h1 s  s=σo +jω

i=2

σ+jω has finitely many unstable zeros where r = e−( N ) . Let r0 is the root of (10). Then, |ro | = e−σ/N , σ = −N ln |ro |. Therefore, the system R has finitely many unstable zeros if and only if all the roots of the polynomial (10) has magnitude greater than one. Note that if there exists a root ro of (10) with |ro | ≤ 1, then there are infinitely many unstable zeros of R converging to |ro | − jN (∠ro + 2πk) as k → ∞ where k ∈ Z ro,k = − lnN 2 and ∠ro is the phase of the complex number ro .

Corollary 2.1. The time-delay system R has finitely many unstable zeros if and only if R is a RTDS or R is a NTDS satisfying Lemma 2.1.

ROCOND'06, Toulouse, France, July 5-7, 2006

A time delay system with finitely many unstable zeros will be called an define the nF -system.−hWe is ¯ R (s)e as R(s) := conjugate of R(s) = i i=1 −hn s e R(−s)MC (s) where MC is inner, finite dimensional whose poles are poles of R. For the above example, we have s + 3 + 2(s − 1)e−0.4s R(s) = (s + 1)2 2  where h1 = 0, h2 = 0.4 and MC (s) = s−1 . So, the s+1 conjugate of R(s) can be written as, 2(s + 1) + (s − 3)e−0.4s ¯ . (11) R(s) = (s + 1)2 ¯ has finitely Corollary 2.2. The time-delay system R many unstable zeros if and only if R is a ATDS or ¯ satisfying Lemma 2.1. R is a NTDS with R ¯ has finitely many The system R whose conjugate R unstable zeros is an I-system. Using Corollary 2.1, an equivalent condition for assumption A.3 is the following. Corollary 2.3. Plant (6) has finitely many unstable zeros or poles if and only if R or T is an F -system. Using Corollary 2.3, it is easy to check whether the plant has finitely many unstable or zeros. After putting the plant in the form (6), if R or T is RTDS, then assumption A.3 is satisfied; if R or T is NTDS and Lemma 2.1 is satisfied at least for one of them, then assumption A.3 holds. It is well known that, since R ∈ H∞ , functions in the form R admit inner outer factorizations (12) R = mn N o where mn is inner and No is outer. To illustrate this first assume that R is an F -system. By Corollary 2.1, it has finitely many unstable zeros. Define an inner function MR whose zeros are unstable zeros of R. Note that MR is finite dimensional, rational function. Then, R can be factorized as in (12) where mn = MR and No = MRR . Note that unstable zeros of R are cancelled by zeros of MR , therefore No is outer and mn is inner by construction of MR . Similarly, if R is an I-system, ¯ has finitely many unstable zeros. By Corollary 2.2, R Define an inner function MR¯ whose zeros are unstable ¯ Using this result, R can be factorized as in zeros of R. ¯ R R (12) where mn = R ¯ and No = M ¯ . ¯ MR R

Corollary 2.4. The plant P = R T satisfies A.3 − A.4 if one of the following conditions are valid: i) R is I-system and T is F -system (IF plant), ii) R is F -system and T is I-system (FI plant), iii) R is F -system and T is F -system (FF plant). Proof. The TDS (6) should have finitely many unstable zeros or poles to apply Skew-Toeplitz approach. By Corollary 2.1, R or T should be a F -system which covers all the cases except R and T are I-systems. Recall that P (6) can be factorized as

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mn,R No,p R mn,R = P = = No T mn,T No,T mn,T N where No = No,R is outer function. Note that when R o,T is F or I-system, mn,R is finite or infinite dimensional respectively. Similarly, when T is F or I-system, mn,T is finite or infinite dimensional respectively. Therefore, the plant (6) can be factorized as (1) or (3). 2 Remarks: (1) By Corollary 2.4, it is easy to check whether assumptions A.3 − A.4 are satisfied or not. For the plant (9) in the example, T is a RTDS, by Corollary 2.1, T is an F -system. R is a NTDS ¯ satisfies Corollary 2.2, therefore, R is a Iand R ¯ (11) is 1 + 1 r and root of system, i.e., ϕ(r) for R 2 the polynomial has magnitude greater than 1. (2) One can show that R or T has infinitely many imaginary-axis zeros if and only if corresponding ϕ(r) has a root with magnitude 1 in Lemma 2.1. Since by assumption A.2, P has no imaginary axis poles or zeros, this possibility is eliminated. In fact, if plant P does not have infinitely many imaginary-axis poles or zeros, the magnitude of roots of ϕ(r) is never equal to 1. (3) For a given system R, if magnitudes of all roots of ϕ(r) in Lemma 2.1 are smaller than one, then R is an I-system. ¯ is an I-system. (4) R is an F -system ⇐⇒ R 2.2 FIR Part of the Time Delay Systems We now show a special structure of time delay systems. This key lemma is used in the next section. Lemma 2.2. Let R be as in Lemma 2.1 and MR be a finite dimensional system whose zeros are included in the zeros of R. Let Sz+ be the set of common C+ zeros of R and MR . Then MRR , can be decomposed as R MR = HR (s) + FR (s), where HR is a system whose poles are outside of Sz+ and the impulse response of FR has finite support (by a slight abuse of notation we say FR is an FIR filter). Proof. For simplicity assume that z1 , z2 , . . . , znz ∈ Sz+ are distinct. We can rewrite MRR as MRR = n Ri −hi s , and decompose each term by partial i=1 MR e Ri fraction, MR = Hi + Fi where the poles of Fi are elements of Sz+ and define the terms HR and FR as, n n   Hi (s)e−hi s , FR (s) = Fi (s)e−hi s . HR (s) = i=1

i=1

where Fi is strictly proper and FR (zk ) is finite ∀ i = 1, . . . , nz . The lemma ends if we can show that FR is FIR filter. Inverse Laplace transform of FR can be written as,    nz n  zk (t−hi ) fR (t) = k=1 e uhi (t) i=1 Res{Fi (s)} s=zk

where uhi (t) = u(t − hi ), u(t) and Res(.) are unit step function and the residue of the function respectively. For t > hn , we have    nz zk t n  −hi zk fR (t) = . e Res{F (s)} e  i k=1 i=1 s=zk

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    Since, Res{Fi (s)} = Ri (zk )Res{MR (s)} , s=z s=zk k     z  ezk t Res{MR (s)} R(zk ) ≡ 0 fR (t) = nk=1 s=zk

z are the zeros of R. for t > hn using the fact {zk }nk=1 Therefore, we can conclude that FR is a FIR filter with support [0, hn ]. Note that the above arguments are also valid for common zeros with multiplicities in Sz+ . 2 Note that this decomposition eliminates unstable pole-zero cancellation in MRR and brings it into a form which is easy for numerical implementation. Lemma 2.2 explains the FIR part of the H∞ controllers as shown below. Assume that R is defined as in Definition 2.1 and R0 is a bi-proper, finite diRi mensional system. By partial fraction, R = Ri,r + 0 Ri,0 ∀i = 1, . . . , n, where the Ri,0 is strictly proper transfer function whose poles are same as the zeros of R0 . Then, the decomposition operator, Φ, is defined as, Φ(R, R0 )= HR + FR n n where HR = i=1 Ri,r e−hi s and FR = i=1 Ri,0 e−hi s are infinite dimensional systems. Note that if the zeros of R0 are also unstable zeros of R, then FR is a FIR filter by Lemma 2.2.

3. MAIN RESULTS In this section, we construct the optimal H∞ controller for the plant P , (6), satisfying assumptions A.1 − A.4. By Corollary 2.4, the plant, P = R T , is assumed to be either IF, FI or FF plant. For each case, we will find optimal H∞ controller and obtain a structure where there is no internal unstable pole-zero cancellation in the controller. 3.1 Factorization of the Plants In order to apply the Skew- Toeplitz approach, we need to factorize the plant as in (1) or (3). 3.1.1. IF Plant Factorization Assume that the plant in (6) satisfies A.1 − A.4, and R is I-system and T is F -system. Then P is in the form (1), where {eh1 s R} m ˆ n = e−(h1 −τ1 )s MR¯ , m ˆ d = MT , ¯ R ¯ MT ˆo = R . (13) N MR¯ {eτ1 s T } where MR¯ is an inner function whose zeros are the ¯ unstable zeros of R(s). Since R is I-system, conjugate of R has finitely many unstable zeros, so MR¯ is welldefined. Similarly, zeros of MT are unstable zeros of T . Note that m ˆ n and m ˆ d are inner functions, infinite and ˆo is an outer term. finite dimensional respectively. N 3.1.2. FI Plant Factorization Let the plant (6) satisfy A.1 − A.4 (with h1 = τ1 = 0), and assume R is F -system and T is I-system. Then the plant P can be factorized as in (3), T ˜o = R MT¯ . ˜ d = MR (s), N m ˜ n = MT¯ ¯ , m MR T¯ T The zeros of MR are right half plane zeros of R. The unstable zeros of T¯ (s) are the same as the zeros of MT¯ . Similar to previous section, conjugate of T has finitely many unstable zeros since T is an I-system.

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The right half plane pole-zero cancellations in m ˜ n and ˜o will be eliminated in Section 3.2.2 by the method N of Section 2.2. 3.1.3. FF Plant Factorization Let P = R/T satisfy A.1 − A.4, with R and T being F -systems. In this case P is in the from (1), m ˆ n = e−(h1 −τ1 )s MR , m ˆ d = MT (s), h1 s ˆo = {e R} MT (14) N MR {eτ1 s T } where MR and MT are inner functions whose zeros are unstable zeros of R and T respectively. Note that when h1 = τ1 = 0, m ˆ n is finite dimensional. Then, exact unstable pole-zero cancellations are possible in ˆo ). this case (except the ones in N 3.2 Optimal H∞ Controller Design Optimal H∞ controllers for problems (2) and (4) are ¨ given in (Toker and Ozbay, 1995) and (G¨ um¨ u¸ssoy and ¨ Ozbay, 2004) for the plants (1) and (3) respectively. Given the plant and the weighting functions, the optimal H∞ cost, γopt can be found as described in these papers. Then, one needs to compute transfer functions labeled as Eγopt , Fγopt and L. Due to space limitations ¨ we skip this procedure, see (Toker and Ozbay, 1995) ¨ and (G¨ um¨ u¸ssoy and Ozbay, 2004) for full details. Instead, we now simplify the structure of the controllers so that a reliable implementation is possible, i.e. there are no internal unstable pole-zero cancellations. 3.2.1. Controller Structure of IF Plants By using the ¨ method in (Toker and Ozbay, 1995; Foias et al., 1996), the optimal controller can be written as,  τs  e 1 T Kγˆopt MT Cˆopt = R¯ (15) + eτ1 s RFγˆopt L MR ¯ where Kγˆopt = Eγˆopt Fγˆopt MT L. In order to obtain this structure of controller: (1) Do the necessary cancellations in Kγˆopt , (2) Partition, Kγˆopt as, Kγˆopt = θγˆopt θT where θγˆopt is a bi-proper transfer function. The zeros of θγˆopt are right half plane zeros of Eγˆopt MT , (3) By Lemma 2.2, obtain (HT , FT ), (HR1 ,FR1 ) and (HR2 , FR2 ) using the partitioning operator, HT + FT = Φ(eτ1 s T θT , MT ), ¯ MR¯ θγˆopt ), HR1 + FR1 = Φ(R, HR2 + FR2 = Φ(eτ1 s RFγˆopt L, θγˆopt ). Then, the optimal controller has the form, HT + FT Cˆopt = (16) Hγˆopt + Fγˆopt where HT , Hγˆopt = HR1 + HR2 are TDS and FT , Fγˆopt = FR1 + FR2 are FIR filters. The controller has no unstable pole-zero cancellations. 3.2.2. Controller Structure of FI Plants After the data transformation is done shown as shown in ¨ (G¨ um¨ u¸ssoy and Ozbay, 2004) γ˜opt , Eγ˜opt , Fγ˜opt and L can be found as in IF plant case. We can write the inverse of the optimal controller similar to (15):

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−1 = C˜opt

Kγ˜opt



R MR



(17) + T Fγ˜opt L where Kγ˜opt = Eγ˜opt Fγ˜opt MR L. Similar to IF plant case, we can obtain a reliable controller structure: (1) Do the necessary cancellations in Kγ˜opt , (2) Partition, Kγ˜opt as, Kγ˜opt = θγ˜opt θR where θγ˜opt is a bi-proper transfer function. The zeros of θγ˜opt are unstable zeros of Eγ˜opt MR , (3) By Lemma 2.2, obtain (HR , FR ), (HT,1 ,FT1 ) and (HT2 , FT2 ) using the partitioning operator, HR + FR = Φ(RθR , MR ), HT1 + FT1 = Φ(T¯, MT¯ θγ˜opt ), HT2 + FT2 = Φ(T Fγ˜opt L, θγ˜opt ). Then, the optimal controller has the form, Hγ˜opt + Fγ˜opt C˜opt = . (18) HR + F R where HR , Hγ˜opt = HT1 + HT2 are TDS and FR , Fγ˜opt = FT1 + FT2 are FIR filters. The controller has no unstable pole-zero cancellations. Note that the optimal controller is dual case of IF plants, R and T are interchanged with h1 = τ1 = 0. T¯ MT¯

3.2.3. Controller Structure of FF Plants Structure of FF plants is similar to that of IF plants. We can calculate γˆopt , Eγˆopt , Fγˆopt , L by the method in (Toker ¨ and Ozbay, 1995; Foias et al., 1996) and write optimal controller as: τ1 T} Kγˆopt {e M T (s) (19) Cˆopt = {eh1 s R} τ1 s RF + e L γ ˆ opt MR where Kγˆopt = Eγˆopt Fγˆopt MT L. The optimal H∞ controller structure can be found by following similar steps as in IF plants. The controller structure will be the same as in (16). Note that when h1 = τ1 = 0, since m ˆ n in (14) is finite dimensional, it possible to cancel the zeros of θγˆopt with denominator. 4. EXAMPLE We consider IF plant (7) and weights as W1 (s) = and W2 (s) = 0.2(s + 1.1). After the plant is factorized as (13), the optimal H∞ cost for two block problem (2) is γˆopt = 0.7203. The impulse responses of FT and Fγˆopt , of the controller (16), are FIR as in Figures 1 and 2, respectively. 2s+2 10s+1

Impulse response of F (s) T

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where Ai ∈ Rn×n , bj ∈ Rn×1 , ck ∈ R1×n and d ∈ R. Define x(t) := [x1 (t), . . . , xn (t)]T . The timeA c b , {hb,i }ni=1 , {hc,i }ni=1 are nonnegative delays, {hA,i }ni=1 rational numbers with ascending ordering respectively and hd ≥ 0. Therefore, we can design an optimal H∞ controller for the plant (20) if there are no imaginary axis poles or zeros (or the weights are chosen in such a way that certain factorizations in (Foias et al., 1996) can be done). In general to see the plant type (IF, FI, FF), the transfer function should be obtained first, then using R and T , one can decide the plant type by Corollary 2.1 and 2.2. The optimal H∞ controller can be found by factorization of the plant and elimination of unstable pole-zero cancellations.

Impulse response of F (s) γ

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Fig. 2. fγˆopt (t) 5. CONCLUDING REMARKS

REFERENCES

In this paper we have discussed general time delay systems and defined FI, IF and FF types of plants. We showed how assumptions of the Skew Toeplitz theory can be checked, and illustrated numerically stable implementations of the optimal H∞ controllers, avoiding internal pole-zero cancellations. We should also mention that if the plant P is written in terms of specific time delay factors, we may still design optimal H∞ controllers even if P is not in any of the types we have considered (i.e. IF, FI, and FF types). It is possible to design H∞ controller for the following cases. Given the plant P = R T , assume that T is an F system and R is neither F or I system, but it can be factorized as R = RF RI where RF is an F system and RI is an I system. Then, we can factorize the plant as (1), RI m ˆ n = MRF MR¯ I ¯ , m ˆ d = MT , RI

¯ RI MT RF ˆ No = MRF MR¯ I T where MRF , MR¯ I and MT are finite dimensional, inner ¯I functions whose zeros are unstable zeros of RF , R and T respectively. By this factorization, the optimal controller can be obtained as in (16). For the dual case, let R be an F system and T = TF TI where TF is an F system and TI is an I system. Now the plant is in the form (3), TI m ˜ n = MR , m ˜ d = MTF MT¯I ¯ , T I





MT¯I MTF R ˜ No = MR TF T¯I where MR , MT¯I and MTF are finite dimensional, inner functions whose zeros are unstable zeros of R, T¯I and TF respectively. Now the optimal controller can be obtained as in (18). Another interesting point to note is that the following plant is a special case of an FF system: nb nA   Ai x(t − hA,i ) + bj u(t − hb,j ), x(t) ˙ = i=1

y(t) =

nc 

k=1

j=1

ck x(t − hc,k ) + du(t − hd )

(20)

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